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# The Børn-Haber Cycle

APCHEM-KLLYER

Consider the following table showing properties of the neutral atoms and most stable ions of four representative elements from the fourth period.

Atom Ionization Energy $\left ( \cfrac{kJ}{mol} \right )$ Electron Affinity $\left ( \cfrac{kJ}{mol} \right )$ Atomic Radius (pm) Most Stable Ion Ionic Radius (pm)
$K$ 419 N/A 227 $K^+$ 133
$Ca$ (1) 589
(2) 1145
N/A 197 $Ca^{2+}$ 99
$S$ N/A (1) $-$201
(2) + 532
104 $S^{2-}$ 184
$Cl$ N/A $-$ 348 99 $Cl^-$ 181

$\$
Note: 1 pm = 10$^{-10}$ cm.

The Børn-Haber cycle is a way to describe the formation of a binary ionic compound starting with a metal and a nonmetal, both in their elementary form. For the case in which an alkali metal reacts with a diatomic halogen (e.g., potassium and chlorine), such a reaction may be represented by a generic equation of the form:

$$M(s)+\frac { 1 }{ 2 } { X }_{ 2 }\rightarrow { M }^{ + }{ X }^{ - }(s)$$

The reaction is envisioned as occurring as a series of steps:

(1) The metal is converted to gaseous atoms.
(2) The gaseous metal atoms are converted to gaseous cations.
(3) The nonmetal is converted to gaseous atoms.
(4) The gaseous nonmetal is converted to gaseous anions.
(5) The gaseous cations and anions unite to form a crystalline ionic solid.

As you see, ionization energies are provided for the metals but not the nonmetals.

Similarly, electron affinities are provided only for the nonmetals, not for the metals.

Based on the information given in the table, and your knowledge of the energies involved, answer the following.

Which of the following accounts for the fact that of all the electron affinities shown, only one is positive?

A

The second electron added to the ${ S }^{ - }$ ion must move to a higher energy level.

B

The first electron added to the gaseous sulfur atom becomes part of the core electrons, causing added repulsion.

C

The second electron added to gaseous sulfur is being added to a particle that already is negatively charged. The electron and the ${ S }^{ - }$ ion repel one another.

D

This is an anomalous behavior; second electron affinities are usually somewhat more exothermic than the first electron affinity, much as the second ionization energy of a metal such as calcium is more endothermic than the first.